Finite Group Databasebeta

← #6501937

Subgroups — #6501937

Order 512.

1782 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2423C211yesmaximal, characteristic
2#34212C211yesmaximal, characteristic
3#28121C212yesmaximal, complemented
4#27681C212yesmaximal, complemented
5#27603C211yesmaximal, characteristic
6#27164C214yesmaximal, complemented
7#26662C214yesmaximal, complemented
8#26559C214yesmaximal, complemented
9#24906C214yesmaximal, complemented
10#24006C211yesmaximal, characteristic
11#24006C211yesmaximal, characteristic
12#23921C211yesmaximal, characteristic
13#22079C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic
15C4×#451C212yesmaximal, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#484C2212yes—
20C2×#484C2212yes—
21C2×#451C2212yes—
22C2×#451C2212yes—
23C2×C4×Q16C2214yescomplemented
24#2423—22no—
25#2423—22no—
26#2423C2214yes—
27#2423C2214yes—
28C2×#399—21no—
29C2×#398—21no—
30#2368—22no—
31#2360C2212yes—
32#2360C2212yes—
33#2345—22no—
34#2345C2212yes—
35Q8×C22⋊C4C2212yes—
36#2329—21no—
37#2329—21no—
38#2315—21no—
39C4×C22⋊Q8C2212yes—
40C4×C22⋊Q8C2212yes—
41C2×#391C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#387—21no—
45C2×#384C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48#2273—21no—
49C2×C4×C22⋊C4C2211yescharacteristic
50#2100C2214yescomplemented
51#2095C2214yescomplemented
52#2076C2214yes—
53#2068C2214yescomplemented
54#2056C2214yescomplemented
55#2036C2214yes—
56#2036C2214yes—
57#2001C2214yes—
58#1947C2214yescomplemented
59#1942C2214yescomplemented
60#1933C2214yescomplemented
61#1928C2214yescomplemented
62#1923C2214yes—
63#1923C2214yes—
64#1921C2214yescomplemented
65#1878C2214yescomplemented
66#1878C2214yescomplemented
67#1872C2212yes—
68#1872C2212yes—
69#1872C2212yes—
70#1872C2212yes—
71#1863C2214yescomplemented
72#1863C2214yescomplemented
73#1852C2212yes—
74#1852C2212yes—
75#1852C2212yes—
76#1852C2212yes—
77#1824C2212yes—
78#1824C2212yes—
79#1803C2214yescomplemented
80#1797C2212yes—
81#1787C2212yes—
82#1787C2212yes—
83#1779C2211yescharacteristic
84#1779C2211yescharacteristic
85C4×#103C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C22×Q16C2314yes—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—44no—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×C4×Q8—22no—
102C2×C4×Q8—44no—
103C2×C4×Q8C2314yes—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4D411yescharacteristic
123C22×C22⋊C4D411yescharacteristic
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#484—28no—
128#484—44no—
129#451—28no—
130#451—44no—
131C4×Q16—48no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C4×C8C2314yes—
149#399—24no—
150#399—24no—
151#398—24no—
152#398—24no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391C2312yes—
157#391C2314yes—
158#391C2312yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#389C2312yes—
164#389C2314yes—
165#388—24no—
166#388—24no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—24no—
174#387—24no—
175#385—22no—
176#385—22no—
177#385—22no—
178#385—22no—
179#384—22no—
180#384C2314yes—
181#380—21no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—22no—
186#380—22no—
187#380C2312yes—
188#380C2312yes—
189#380C2312yes—
190#380C2312yes—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4C2312yes—
204C4×C22⋊C4C2312yes—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2314yes—
219#336C2314yes—
220C25C2×D411yescharacteristic
221C22×Q8—24no—
222C22×Q8—24no—
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—24no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8—42no—
229C22×Q8C2412yes—
230C22×Q8C2412yes—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—24no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×Q16—28no—
253C2×Q16—28no—
254C2×Q16—44no—
255C22×C8C2414yes—
256C22×C8C2414yes—
257C22⋊Q8—28no—
258C22⋊Q8—28no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—42no—
265C22⋊Q8—42no—
266C22⋊Q8—44no—
267C22⋊Q8—44no—
268C22⋊Q8#4118yes—
269C4×Q8—28no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—48no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4Q1618yescomplemented
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—44no—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340#107—28no—
341#107—44no—
342#107—44no—
343C4⋊C8—44no—
344#103—28no—
345#103—28no—
346#103—28no—
347#103—44no—
348#103—44no—
349#103—44no—
350C22⋊C8—24no—
351C22⋊C8—24no—
352C22⋊C8—42no—
353C22⋊C8—42no—
354C22⋊C8—42no—
355C22⋊C8—42no—
356C4×C8—44no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376C24—22no—
377C24—22no—
378C24—41no—
379C24—41no—
380C24—41no—
381C24—41no—
382C24C2×Q1614yes—
383C24C22×D411yescharacteristic
384C24C22×D411yescharacteristic
385C24C22×D411yescharacteristic
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—416no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—81no—
398C2×Q8—81no—
399C2×Q8—88no—
400C2×Q8C2×#4114yes—
401C2×Q8H(5,𝔽2)14yes—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—88no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—81no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4C2×Q1614yes—
463C22×C4C2×Q1614yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C2×#4114yes—
472C22×C4C2511yescharacteristic
473Q16—416no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C4⋊C4—24no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4C2×#4114yes—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4C2×Q1618yes—
517C22⋊C4C2×Q1618yes—
518C42—24no—
519C42—24no—
520C42—42no—
521C42—42no—
522C42—48no—
523C42—82no—
524C42—82no—
525C42—82no—
526C42—82no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—28no—
544C23—44no—
545C23—44no—
546C23—82no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23#52512yes—
551C23C22×Q1612yes—
552C23C22×Q1612yes—
553C23C22×Q1612yes—
554C23#57512yes—
555C23C23×D411yescharacteristic
556Q8—28no—
557Q8—28no—
558Q8—48no—
559Q8—44no—
560Q8—44no—
561Q8—84no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—162no—
621C2×C4—162no—
622C2×C4#52512yes—
623C2×C4C22×Q1614yes—
624C2×C4C22×Q1614yes—
625C2×C4C22×#4112yes—
626C2×C4C2×H(5,𝔽2)12yes—
627C2×C4#58512yes—
628C8—44no—
629C8—44no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—88no—
651C22#310414yes—
652C22C2×#52511yescharacteristic
653C22#348411yescharacteristic
654C22#348511yescharacteristic
655C22C23×Q1611yescharacteristic
656C22C2×#57511yescharacteristic
657C22#358511yescharacteristic
658C22#358611yescharacteristic
659C4—24no—
660C4—22no—
661C4—24no—
662C4—42no—
663C4—41no—
664C4—41no—
665C4—44no—
666C4—44no—
667C4—42no—
668C4—44no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4#351314yes—
675C2—24no—
676C2—42no—
677C2—42no—
678C2#3535411yescharacteristic
679C2C2×#310411yescharacteristic
680C2#6056711yescharacteristic
681C2#6156311yescharacteristic
682C2#6157711yescharacteristic
683C2#6169811yescharacteristic
684C2#6171611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #28121C2semidirect
4. #27681C2semidirect
6. #27164C2semidirect
7. #26662C2semidirect
8. #26559C2semidirect
9. #24906C2semidirect
13. #22079C2semidirect
15. C4×#451C2semidirect
23. C2×C4×Q16C22semidirect
50. #2100C22semidirect
51. #2095C22semidirect
53. #2068C22semidirect
54. #2056C22semidirect
58. #1947C22semidirect
59. #1942C22semidirect
60. #1933C22semidirect
61. #1928C22semidirect
64. #1921C22semidirect
65. #1878C22semidirect
66. #1878C22semidirect
71. #1863C22semidirect
72. #1863C22semidirect
79. #1803C22semidirect
85. C4×#103C22semidirect
323. C2×C22⋊C4Q16semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#24232
#342122
#281212
#276812
#276032
#271642
#266622
#265592
#249062
#240062
#239212
#220792
#219022
C4×#4512